Define the determinant of an $n \times n$ matrix $A$, and prove from your definition that if $A^{\prime}$ is obtained from $A$ by an elementary row operation (i.e. by adding a scalar multiple of the $i$ th row of $A$ to the $j$ th row, for some $j \neq i$ ), then $\operatorname{det} A^{\prime}=\operatorname{det} A$.

Prove also that if $X$ is a $2 n \times 2 n$ matrix of the form

$$\left(\begin{array}{ll} A & B \\ O & C \end{array}\right)$$

where $O$ denotes the $n \times n$ zero matrix, then $\operatorname{det} X=\operatorname{det} A$ det $C$. Explain briefly how the $2 n \times 2 n$ matrix

$$\left(\begin{array}{ll} B & I \\ O & A \end{array}\right)$$

can be transformed into the matrix

$$\left(\begin{array}{cc} B & I \\ -A B & O \end{array}\right)$$

by a sequence of elementary row operations. Hence or otherwise prove that $\operatorname{det} A B=$ $\operatorname{det} A \operatorname{det} B$.